To transform the graph of the function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] to represent the function [tex]\( g(x) = \sqrt[3]{x} - 3 \)[/tex], we follow these steps:
1. Understand the Basic Function [tex]\( f(x) = \sqrt[3]{x} \)[/tex]:
- The function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] represents the cube root of [tex]\( x \)[/tex].
2. Determine the Transformation:
- The function [tex]\( g(x) = \sqrt[3]{x} - 3 \)[/tex] is obtained by translating the graph of [tex]\( f(x) \)[/tex] down by 3 units. This means that for any point [tex]\((a, \sqrt[3]{a})\)[/tex] on the graph of [tex]\( f(x) \)[/tex], the corresponding point on the graph of [tex]\( g(x) \)[/tex] will be [tex]\((a, \sqrt[3]{a} - 3)\)[/tex].
3. Choose a Specific Point on [tex]\( f(x) \)[/tex] to Illustrate the Transformation:
- Consider a specific point on the graph of [tex]\( f(x) \)[/tex]. For illustration, let's choose [tex]\( x = 8 \)[/tex].
- For [tex]\( f(x) = \sqrt[3]{x} \)[/tex], when [tex]\( x = 8 \)[/tex], we calculate [tex]\( y \)[/tex]:
[tex]\[
y = \sqrt[3]{8} = 2
\][/tex]
- Hence, the point [tex]\((8, 2)\)[/tex] is on the graph of [tex]\( f(x) \)[/tex].
4. Apply the Translation to Find the Corresponding Point on [tex]\( g(x) \)[/tex]:
- We need to move the point [tex]\((8, 2)\)[/tex] down by 3 units to find the corresponding point on [tex]\( g(x) \)[/tex].
- This means the new [tex]\( y \)[/tex]-coordinate will be [tex]\( 2 - 3 = -1 \)[/tex].
Therefore, the point [tex]\((8, -1)\)[/tex] is on the graph of [tex]\( g(x) \)[/tex].
To summarize the transformation and fill in the blanks:
To transform the graph of [tex]\( f(x) = \sqrt[3]{x} \)[/tex] to represent the function [tex]\( g(x) = \sqrt[3]{x} - 3 \)[/tex], translate the graph of [tex]\( f(x) \)[/tex] down [tex]\( 3 \)[/tex] units. The point [tex]\((8, -1)\)[/tex] is on the graph of [tex]\( g(x) \)[/tex].
